The Allen-Cahn equation on closed manifolds
Abstract
We study global variational properties of the space of solutions to $-\varepsilon^2\Delta u + W'(u)=0$ on any closed Riemannian manifold $M$. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min-max and have index 1. We show that if $\varepsilon$ is not small enough, in terms of the Cheeger constant of $M$, then there are no interesting solutions. However, we show that the number of min-max solutions to the equation above goes to infinity as $\varepsilon\to 0$ and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed $\varepsilon$, as shown recently by G. Smith. We also show that the energy of the min-max solutions accumulate, as $\varepsilon \to 0$, around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with ${\rm Ric}_M>0$, the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet-Rosenberg. Finally, we prove that the min-max energy values are bounded from below by the widths of the area functional as defined by Marques-Neves.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2016
- DOI:
- 10.48550/arXiv.1608.06575
- arXiv:
- arXiv:1608.06575
- Bibcode:
- 2016arXiv160806575G
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Analysis of PDEs
- E-Print:
- 46 pages